Ordinal Preference: Game-Theoretic Aspects of Computational Social Choice

1. Ordinal Preference Representation and AggregationGame-Theoretic and Combinatorial Aspects of Computational Social Choice Lirong Xia EPFL June 15, 2012

2. Preference Aggregation: Social Choice > > voting rule > > > >

3. Social Choice and Computer Science Computational thinking + optimization algorithms CS Social Choice 21th Century Strategic thinking + methods/principles of aggregation PLATO 4thC. B.C. LULL 13thC. BORDA 18thC. CONDORCET 18thC. ARROW 20thC. TURING et al. 20thC. PLATO et al. 4thC. B.C.---20thC.

4. Many applications • People/agents often have conflicting preferences, yet they have to make a joint decision

5. Applications • Multi-agent systems [Ephrati and Rosenschein91] • Recommendation systems [Ghoshet al. 99] • Meta-search engines [Dwork et al. 01] • Belief merging [Everaereet al. 07] • Human computation (crowdsourcing) • etc.

6. A burgeoning area • Recently has been drawing a lot of attention • IJCAI-11: 15 papers, best paper • AAAI-11: 6 papers, best paper • AAMAS-11: 10 full papers, best paper runner-up • AAMAS-12 9 full papers, best student paper • EC-12: 3 papers • Workshop: COMSOC Workshop 06, 08, 10, 12 • Courses taught at Technical University Munich (Felix Brandt), Harvard (Yiling Chen), U. of Amsterdam (UlleEndriss)

7. Outline 1. Game-theoretic aspects NP- Hard 2. Combinatorial voting NP- Hard

8. Common voting rules(what has been done in the past two centuries) • Mathematically, a voting rule is a mapping from {All profiles} to {outcomes} • an outcome is usually a winner, a set of winners, or a ranking • m : number of alternatives (candidates) • n : number of voters • Positional scoring rules • A score vectors1,...,sm • For each vote V, the alternative ranked in the i-th position gets si points • The alternative with the most total points is the winner • Special cases • Borda, with score vector (m-1, m-2, …,0) • Plurality, with score vector (1,0,…,0) [Used in the US]

9. An example • Three alternatives {c1, c2, c3} • Score vector(2,1,0) (=Borda) • 3 votes, • c1 gets 2+1+1=4, c2 gets 1+2+0=3, c3 gets 0+0+2=2 • The winner is c1 2 1 0 2 1 0 2 1 0

10. Single transferable vote (STV) • Also called instant run-off voting or alternative vote • The election has m-1rounds, in each round, • The alternative with the lowest plurality score drops out, and is removed from all of the votes • The last-remaining alternative is the winner • [used in Australia and Ireland] a > b > c > d a > c > d a > c a > c c > d > a c > d > a >b c > a c > a • d > a > b > c • d > a > c • b > c > d >a • c > d >a a

11. Strategic voters • Manipulation: a voter (manipulator) casts a vote that does not represent her true preferences, to make herself better off • A voting rule is strategy-proof if there is never a (beneficial) manipulation under this rule • How important strategy-proofness is as an desired axiomatic property? • compared to other axiomatic properties

12. Manipulation under plurality rule (ties are broken in favor of ) > > > > Plurality rule > > > >

13. Any strategy-proof voting rule? • No reasonable voting rule is strategyproof • Gibbard-Satterthwaite Theorem [Gibbard Econometrica-73, Satterthwaite JET-75]:When there are at least three alternatives, no voting rules except dictatorships satisfy • non-imposition: every alternative wins for some profile • unrestricted domain: voters can use any linear order as their votes • strategy-proofness • Axiomatic characterization for dictatorships!

14. Computational thinking • Use a voting rule that is too complicated so that nobody can easily figure out who will be the winner • Dodgson: computing the winner is -complete [Hemaspaandra, Hemaspaandra, &Rothe JACM-97] • Kemeny: computing the winner is NP-hard [Bartholdi, Tovey, &Trick SCW-89]and -complete [Hemaspaandra, Spakowski, & Vogel TCS-05] • The randomized voting rule used in Venice Republic for more than 500 years [Walsh&Xia AAMAS-12] • We want a voting rule where • Winner determination is easy • Manipulation is hard

15. Overview Manipulation is inevitable (Gibbard-Satterthwaite Theorem) Can we use computational complexity as a barrier? Why prevent manipulation? • Yes • May lead to very • undesirable outcomes Is it a strong barrier? • No How often? Other barriers? • Seems not very often • Limited information

16. Manipulation: A computational complexity perspective If it is computationallytoo hard for a manipulator to compute a manipulation, she is best off voting truthfully • Similar as in cryptography For which common voting rules manipulation is computationally hard? NP- Hard

17. Unweightedcoalitional manipulation (UCM) problem • Given • The voting rule r • The non-manipulators’ profile PNM • The number of manipulators n’ • The alternative c preferred by the manipulators • We are asked whether or not there exists a profile PM (of the manipulators) such that c is the winner of PNM∪PM under r

18. The stunningly big table for UCM My work

19. What can we conclude? • For some common voting rules, computational complexity provides some protection against manipulation • Is computational complexity a strong barrier? • NP-hardness is a worst-case concept

20. Probably NOT a strong barrier Frequency of manipulability Easiness of Approximation Quantitative G-S

21. An approximation viewpoint • Unweighted coalitional optimization (UCO):compute the smallest number of manipulators that can make cwin • A greedy algorithm has additive error no more than 1 for Borda[Zuckerman, Procaccia, &Rosenschein AIJ-09]

22. An approximation algorithm for positional scoring rules [Xia,Conitzer,& Procaccia EC-10] • A polynomial-time approximation algorithm that works for all positional scoring rules • Additive error is no more than m-2 • Computational complexity is not a strong barrier against manipulation • The cost of successful manipulation can be easily approximated (for some rules)

23. A class of scheduling problems Q|pmtn|Cmax • m* parallel uniform machines M1,…,Mm* • Machine i’s speed is si (the amount of work done in unit time) • n* jobs J1,…,Jn* • preemption: jobs are allowed to be interrupted (and resume later maybe on another machine) • We are asked to compute the minimum makespan • the minimum time to complete all jobs

24. Thinking about UCOpos • Let p,p1,…,pm-1 be the total points that c,c1,…,cm-1 obtain in the non-manipulators’ profile V1 PNM ∪{V1=[c>c1>c2>c3]} = c c p ∨ c1 c1 s1=s1-s2 (J1) p1 p p1 -p p1 –p-(s1-s2) s1-s2 ∨ c3 c2 (J2) s2=s1-s3 s1-s3 p p2 p2 -p p2 –p-(s1-s4) ∨ c2 c3 (J3) p p3 p3 -p p3 –p-(s1-s3) s3=s1-s4 s1-s4

25. The algorithm in a nutshell Scheduling problem Original UCO No more than OPT+m-2 [Gonzalez&Sahni JACM 78] Solution to the UCO Solution to the scheduling problem Rounding

26. Helps to prove complexity of UCM for Borda • Manipulation of positional scoring rules = scheduling (preemptions only allowed at integer time points) • Borda manipulation corresponds to scheduling where the machines speeds are m-1, m-2, …, 0 • NP-hard [Yu, Hoogeveen, & Lenstra J.Scheduling 2004] • UCM for Borda is NP-C for two manipulators • [Davies et al. AAAI-11 best paper] • [Betzler, Niedermeier, & Woeginger IJCAI-11 bestpaper]

27. Next step • The first attempt seems to fail • Can we obtain positive results for a restricted setting? • The manipulators has complete information about the non-manipulators’ votes

28. Information constraints[Conitzer,Walsh,&Xia AAAI-11] • Limiting the manipulator’s information can make dominating manipulation computationally harder, or even impossible

29. Overview • Manipulation is inevitable • (Gibbard-Satterthwaite Theorem) Can we use computational complexity as a barrier? Why prevent manipulation? • Yes • May lead to very • undesirable outcomes Is it a strong barrier? • No How often? Other barriers? • Seems not very often • Limited information

30. Research questions • How to predict the outcome? • Game theory • How to evaluate the outcome? • Price of anarchy [Koutsoupias&Papadimitriou STACS-99] • Not very applicable in the social choice setting • Equilibrium selection problem • Social welfare is not well defined Optimal welfare when agents are truthful Worst welfare when agents are fully strategic

31. Simultaneous-move voting games • Players: Voters 1,…,n • Strategies / reports:Linear orders over alternatives • Preferences:Linear orders over alternatives • Rule:r(P’), where P’ is the reported profile

32. Equilibrium selection problem > > > > Plurality rule > > > > > > > >

33. Stackelberg voting games[Xia&Conitzer AAAI-10] • Voters vote sequentially and strategically • voter 1 → voter 2 → voter 3 → … → voter n • any terminal state is associated with the winner under rule r • At any stage, the current voter knows • the order of voters • previous voters’ votes • true preferences of the later voters (complete information) • rule r used in the end to select the winner • Called a Stackelberg voting game • Unique winner in SPNE (not unique SPNE) • Similar setting in [Desmedt&Elkind EC-10]

34. General paradoxes (ordinal PoA) • Theorem.For any voting rule r that satisfies majority consistency and any n, there exists an n-profile P such that: • (many voters are miserable)SGr(P) is ranked somewhere in the bottom two positions in the true preferences of n-2voters • (almost Condorcet loser) SGr(P) loses to all but one alternative in pairwise elections • Strategic behavior of the voters is extremely harmful in the worst case

35. Food for thought • The problem is still open! • Shown to be connected to integer factorization [Hemaspaandra, Hemaspaandra, & Menton Arxiv-12] • What is the role of computational complexity in analyzing human/self-interested agents’ behavior? • NP-hardness might not be a good answer, but it can be seen as a desired “axiomatic” property • Explore information assumption • In general, why do we want to prevent strategic behavior? • Practical ways to protect election

36. Outline 1. Game-theoretic aspects NP- Hard 2. Combinatorial voting NP- Hard

37. Settings with too many alternatives • Representation/communication: How do voters communicate theirpreferences? • Computation: How do we efficiently compute the outcome given the votes? NP- Hard

38. Combinatorial domains(Multi-issue domains) • The set of alternatives can be uniquely characterized by multiple issues • Let I={x1,...,xp} be the set of p issues • Let Di be the set of values that the i-th issue can take, then C=D1×... ×Dp • Example: • Issues={ Main course, Wine } • Alternatives={} ×{ }

39. Example: joint plan [Brams, Kilgour & Zwicker SCW 98] • The citizens of LA county vote to directly determine a government plan • Plan composed of multiple sub-plans for several issues • E.g., • # of alternatives is exponential in the # of issues

40. Overview Combinatorial voting • New criteria used • to evaluate rules • Strategic considerations • An example of • voting language/rule • Compare new approaches • to existing ones

41. Criteria for combinatorial voting • Criteria for the voting language • Compactness • Expressiveness • Usability: how comfortable voters are about it • Informativeness: how much information is contained • Criteria for the voting rule • Computational efficiency • Whether it satisfies desirable axiomatic properties

42. CP-net [Boutilier et al. JAIR-04] • An CP-net consists of • A set of variablesx1,...,xp, taking values on D1,...,Dp • A directed graph G over x1,...,xp • Conditional preference tables (CPTs) indicating the conditional preferences over xi, given the values of its parents in G • c.f. Bayesian network • Conditional probability tables • A BN models a probability distribution, a CP-net models a partial order

43. CP-nets: An example Variables:x,y,z. Graph CPTs This CP-net encodes the following partial order: x y z

44. Sequential voting rules [Lang IJCAI-07, Lang&XiaMSS-09] • Issues: main course, wine • Order: main course > wine • Local rules are majority rules • V1: > , : > , : > • V2: > , : > , : > • V3: > , : > , : > • Step 1: • Step 2: given , is the winner for wine • Winner: ( , )

45. Previous approaches We want a balanced rule!

46. Sequential voting vs. issue-by-issue voting Acyclic CP-nets (compatible with the same ordering)

47. Yet another approach H-composition vs.Sequential rules

48. AI may help! • Computing local/global Condorcet winner • CSP with cardinality constraints [Li, Vo, & Kowalczyk AAMAS-11] • Applying common voting rules (including Borda) to preferences represented by lexicographic preference trees • Weighted MAXSAT solver [Lang, Mengin, & Xia CP-12]

49. Overview Combinatorial voting • New criteria used • to evaluate rules • Strategic considerations • An example of • voting language/rule • Compare new approaches • to existing ones

50. Strategic consideration • When voters are strategic • how to evaluate the harm? • how to prevent strategic behavior?